Rolling without slipping combines translational motion of the CM and rotation about the CM simultaneously.

The Rolling Constraint: $v_{cm} = \omega R \quad \text{and} \quad a_{cm} = \alpha R$

This constraint means the contact point has zero velocity: $v_{contact} = v_{cm} - \omega R = 0$

Velocity at various points:

Instantaneous axis of rotation: The contact point acts as the instantaneous pivot. The entire rolling body rotates about this point with angular velocity $\omega$.

Energy: $KE_{total} = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I\omega^2 = \frac{1}{2}mv_{cm}^2\!\left(1 + \frac{K^2}{R^2}\right)$

Friction in rolling: Static friction provides the torque for angular acceleration but does zero work (contact velocity = 0).

Condition for rolling without slipping: $f \leq \mu_s mg\cos\theta$. If the required friction exceeds $\mu_s mg\cos\theta$, the body slips.